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Questions tagged [harmonic-analysis]

Harmonic analysis is the generalisation of Fourier analysis. Use this tag for analysis on locally compact groups (e.g. Pontryagin duality), eigenvalues of the Laplacian on compact manifolds or graphs, and the abstract study of Fourier transform on Euclidean spaces (singular integrals, Littlewood-Paley theory, etc). Use the (wavelets) tag for questions on wavelets, and the (fourier-analysis) for more elementary topics in Fourier theory.

Harmonic analysis can be considered a generalisation of Fourier analysis that has interactions with fields as diverse as isoperimetric inequalities, manifolds and number theory.

Abstract harmonic analysis is concerned with the study of harmonic analysis on topological groups, whereas Euclidean harmonic analysis deals with the study of the properties of the Fourier transform on $\mathbb{R}^n$.

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What do modern-day analysts actually do?

In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about the classification of finite simple groups, and…
Jesse Madnick
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Why do we use trig functions in Fourier transforms, and not other periodic functions?

Why, when we perform Fourier transforms/decompositions, do we use sine/cosine waves (or more generally complex exponentials) and not other periodic functions? I understand that they form a complete basis set of functions (although I don't understand…
user6873235
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Motivation for abstract harmonic analysis

I am reading Folland's A Course in Abstract Harmonic Analysis and find this book extremely exciting. However, it seems Folland does not give many examples to illustrate the motivation behind much of the theory. Thus, I wonder whether there is…
Hui Yu
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Prove or disprove a claim related to $L^p$ space

The following question is just a toy model: Let $f:[0,1] \rightarrow \mathbb{R}$ be Lebesgue integrable, and suppose that for any $0\le a
student
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If $f$ is a smooth real valued function on real line such that $f'(0)=1$ and $|f^{(n)} (x)|$ is uniformly bounded by $1$ , then $f(x)=\sin x$?

Let $f : \mathbb R \to \mathbb R$ be a smooth ( infinitely differentiable everywhere ) function such that $f '(0)=1$ and $|f^{(n)} (x)| \le 1 , \forall x \in \mathbb R , \forall n \ge 0$ ( as usual denoting $f^{(0)}(x):=f(x)$) ; then is it true…
user228168
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Learning algebra and harmonic analysis

I've revised my question a bit in response to the (very helpful) advice so far-- I have an engineering background but am interested in learning abstract harmonic analysis. My interest is rather unstructured; the Fourier transform sparked my interest…
Albert
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What is the goal of harmonic analysis?

I am taking a basic course in harmonic analysis right now. Going in, I thought it was about generalizing Fourier transform / series: finding an alternative representation of some function where something works out nicer than it did before. Now,…
fubal
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When does $\lim_{n\to\infty}f(x+\frac{1}{n})=f(x)$ a.e. fail

We know that if $f\in L^1(\mathbb{R})$, then $\|f(\cdot+1/n)-f(\cdot)\|_{L^1}\to 0$ as $n\to \infty$, which implies that there exists a subsequence $f_{n_k}=f(x+\frac{1}{n_k})$ such that $f_{n_k}\to f$ a.e. My problem is that Is there any $f\in…
Yuhang
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A property of exponential of operators

Let $X$ be a Banach space. $A\in B(X)$ is a bounded operator. we can define $e^{tA}$ by $$e^{tA}=\sum_{k=0}^{+\infty}\frac{t^kA^k}{k!}$$ I am interested in this property: If $x\in X$, such that the function $t\mapsto e^{tA}x$ is bounded on…
user165633
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Properties of Haar measure

Let $G$ be a locally compact group (but not discrete) and let $m$ be its left Haar measure. Is it true that $\forall \epsilon$ $\exists$ $C$ such that $C$ is a compact neighborhood of the identity and the measure of $C$ is less than$\epsilon$ ? I…
Grasshopper
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Are Fourier Analysis and Harmonic Analysis the same subject?

Are Fourier Analysis and Harmonic Analysis the same subject? I believe that they are not the same. Maybe there is big difference between those subjects but I need to know what is the main difference between those subjects and what is the main…
IremadzeArchil19910311
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A net version of dominated convergence?

Let $G$ be a locally compact Hausdorff Abelian topological group. Let $\mu$ be a Haar measure on $G$, i.e. a regular translation invariant measure. Let $f$ be fixed in $\mathcal{L}^1(G, \mu)$. Define the function from $G$ to $\mathcal{L}^1(G)$…
Jeff
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Criteria for swapping integration and summation order

I have a function (a potential from an electrostatic potential via a Fourier series) in the form of $$V(x, y, z)=\sum_n\sum_m \ a(x, n, m) b(y, n) c(z, m) \int\int f(u, v) d(u,n) e(v,m) du\, dv$$ Here functions $a(), b(), c(), d(), e()$ are known,…
Dave Terr
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How to show convolution of an $L^p$ function and a Schwartz function is a Schwartz function

We have the Schwartz space $\mathcal{S}$ of $C^\infty(\mathbb{R^n})$ functions $h$ such that $(1+|x|^m)|\partial^\alpha h(x)|$ is bounded for all $m \in \mathbb{N_0}$ and all multi-indices $\alpha$. We are given an $f \in L^p$ with $1\leqslant p <…
nullUser
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Lower bound for the Hardy-Littlewood maximal function implies it is not integrable

I am working on the following problem from Stein and Shakarchi: Let $f$ be an integral function on $\mathbb{R}^d$ such that $\|f\|_{L^1} = 1$ and let $f^*$ by the Hardy-Littlewood maximal function corresponding of $f$. Prove that if $f$ is…
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